Here is an updated version of the ALS_65 shown in the post of November 3, 2020. It’s superimposed on the AIC segments left by the AIC building of last week. It confirmed N3, a step taken by a solver’s complex AIC in the November 10 post, but after the ALS_65. The confirmation replaced the ALS in r1 by a simpler one.

Let’s review the ALS_XZ and the problem of finding ALS.

An ALS is a set of *n* cells in a unit(house) containing candidates of *n *+ 1 values.

ALS_XZ is based on the fact that when a value is removed from an ALS, there are *n* values for *n* cells. The remaining values are *locked*. Every remaining value group in the ALS has a true candidate. In this example, the single 6’s form the grouped weak link between ALS known as the *restricted common*. It guarantees that only one of the two ALS will have a X = 6 in the solution. One of the two value groups of Z = 5 will be locked. One 5-candidate outside the two ALS, 5r8c2, sees both 5 groups, and a true 5 candidate.

ALS_XZ is a consistent source of candidate removals, but spotting them is difficult. Finding them exhaustively seems overwhelmingly so, because ALS are so numerous, and because you have to see the X and Z connections between two ALS for each one. But like AIC building, there is a practical way for human solvers to do it. With an ALS map.

Here is the formidable looking Sysudoku ALS map of ultrahardcore 311, when the ALS_65 above is discovered.

It is the line marked grid, with ALS cells marked by curves. Bi-value cell ALS are already marked in green. Extra breakout panels and colors help make it barely possible to distinguish the ALS of UHC 311. A small circle is color matched to the encircling curve to mark the single candidate values. Trace how the ALS_65 above is spotted, and formulate your rules for finding more of the ALS_XZ.

You invest the time to build this map once, then bring updated copies along with you. ©Powerpoint is good for this. Anticipate using the map for spotting Beeby’s ALS-wings as well. But more generally, remember that after AIC building, we have segments of AIC on the grid that have not been extended by AIC nodes. An ALS node has strong links that exist between the value groups in every ALS. That can mean a second wave of AIC building. We will look into these possibilities in the next post.

So with a little exploration of the map, and with these prospects ahead of us, let’s take the time to walk through the construction of an ALS map. To even talk about it, we must have a way to describe particular ALS. We’ll use cell/value *susets*, a concept used many times in this blog. In the suset, two digit strings identify cell positions and values. The blue ALS is r1 27/256, with cells 2 and 7 of r1 holding 2, 5 and 6 value candidates.

For the map, we need to list all of the ALS in every row, column and box. Let’s start with r1 above, listing each cell by suset:

2/25, 5/56, 7/26

The bv are ALS, and there is an overlap of one value in each of 3 combinations, for

25/256, 27/256, and 57/256. A combination is a union of values and a union of cells.

All combinations now add a cell, reaching 257/256 which includes 3 cells and 3 values, and is not an ALS.

For a more typical case, let’s take the ALS in column 1 above. Here are the susets,

2/2358 4/23489 5/2459 7/589 8/58 9/259

Of course 8/58 is an ALS. Adding value 9 we include cell 7 for an ALS 78/589. Now looking for another such overlap of values, we combine cells 5 and 9 for 59/2459. And then we see 28/2358. In both cases cells are gaining on values. In our new list, we keep only the susets with added values in our revised list,

28/2358 4/23489 59/2459 78/589 8/58 9/259

Notice that the values in 4/23489 will pick up many cells if it has a 5, and no cells if it doesn’t. That gives us 245789/234589. Oops, 6 cells and all 6 values.

Now where is there more value overlap among values to pick up more cells per value? The last three. Add 2 to 589 gets 789/2589. Its an ALS adding one cell and one value added to an ALS. We now see that adding the 4 value to ALS 789/2589 is going to bring in cell 5, another ALS. Let’s list our ALS in order of increasing values:

8/58 78/589 789/2589 5789/24589, leaving behind 28/2358 and 59/2459 .

For homework, how about finding the other ALS in row 8, without checking on the map.

To prepare a map, record the susets for cells and ALS separately rows, columns, and boxes. Here are the ALS tables for UHC 311:

As you place ALS curves on the map, you can mark them for ALS_XZ partnering. To be an ALS_XZ partner, an ALS must have, for both X and for Z:

A single candidate value, or

An aligned value group with a possible Z victim, or

A box confined value group.

In ALS map, I mark single candidates for X and Z with small circles, matched in color with the ALS boundary curve. On the map, our r1 ALS has two single values 5 and 6.

Here, I use small pentagons to mark some potential Z victims of aligned value groups. Our ALS_65 r8 ALS has a matching 6 single value, and this map has the pentagon, but normally victims are not marked, because you can’t anticipate the partnering ALS.

This example of a box confined value group is from French Su-Doku Maestro magazine of July-Sept., 2009, and its L’attaque du cobra example 4. The green ALS , allows the grouped restricted common X = 5. The blue ALS qualifies for the map with the single 5 and the aligned victim 2.

The box confined value group is based on properties of both ALS. In this example, the 5r3c2 single value could be the starting signal. The spotting rule would be to check for an ALS group confined in the box of the single.

This post shows that exhaustive ALS map making can require a huge effort. At least you only have to make the map once, then update it, along with the bv map, X-panel and AIC segments, for an added dimension in solving very hard puzzles. Next week, we apply the map to the UHC 311 case to illustrate the several ways it can be used.

]]>AIC building is a phase of Sysudoku Advanced which follows the bv scan for Sue de Coq, WXY-wings, and XY-chains, and the X-panel analysis for X-chains, grouped X-chains, regular and finned fish, and edge limited freeform pattern analysis. After these systematic and exhaustive elimination methods, the human solver can reasonably begin AIC building, a systematic search for Alternate Inference Chains, with its many possible starting points, continuations, and terminations. Let’s look back to the post of last November 3, where solvers Sudokuwiki and Beeby immediately followed line marking.

A series of four AIC’s resulted, before the first ALS_XZ. With solvers you never see the many trial chains that do not lead to any result. Is it an overwhelming number, though? Maybe not. Consider how the chains generated on a grid are interdependent, restricted as they are by Sudoku placement rules. In the review, three of the four solver AIC start on the same slink. Many segments are shared among these four.

If we set out to find every grouped, fully extended AIC without ALS nodes or 1-way branches, how many would there be? We’ll do our top down, left to right scan for starting slinks. Eligible slinks have a wink directly to a next slink, and we can go off each end of our starting slink. Record your guess on how many distinct AIC, counting extensions and not counting extensions.

Starting off with the r1 5-slink, we can’t leave from 5r1c2 because it doesn’t see another slink partner. In that sense, AIC building is like lite tree building. It’s a surprise how far the AIC goes, getting into a loop but branching out to get to two possible slink terminals, but on 3, not 5.

Veteran readers know that a loop can often start a coloring cluster.

Here we realize the AIC contains a 1-way from 3r1c2. Either terminal 3 sees the victim, and if either is false, the chain makes 5 true in the victim’s cell. In the review, Beeby made the same removal with a complex 1-way, later.

Now we shift the 3’s to mark the new slinks in NW and r1, and move on to the r1c4 slink for an AIC starter. From the 3 it gets into the same loop, but going in the wrong direction to slink out on the 9. From 2r1c4 there’s a long branching AIC, with one branch reaching an ANL.

The second elimination in r1 generates N3, which destroys an ALS_XZ partnering ALS in the review.

Now moving on to the N 4-slink and 4r3c4, we’ll blue out past segments to put them in the background, going to black for new extensions. We get an ANL removal with each extension.

The right branch brings us to an extension to 4r8c9 which teams with our staring 4r3c6 in an ANL removing 4r8c6. Btanching off at 8r8c6 to 4rc5 removes 4r4c6.

Recycling AIC is good.

We get to do it again with a grouped extension from 5r8c1. The prior removals provide that closing slink for a confirming ANL and N4.

I don’t see anything left. A benefit of systematic AIC building is to have many AIC segments on a background layer, for possible use later.

We’ve accounted for all the solver AIC findings. How many AIC did we have to generate? I’d say a reasonably small number.

Next week, we start a similar reverse engineering on the solvers’ ALS findings, introducing a graphics tool for systematic searching. That too, is a complexity nightmare for human searching that can be kept in bounds. The prospect is a systematic order of battle against very difficult puzzles.

]]>“UHC 311 wishes to settle before going to trial.”

“Starting from the toxic orphan, in c2, 8(6,2) is light orange. 2(6,2) immediately vanishes in a puff of logic.”

*Yes, 2r6c2 goes, because either red or orange is true.*

“Also, we have a bridge between orange and green. Red and blue make a toxic pair. 5(9,3) storms off in a huff. “

“Light red can also grow, and 6(7,8) goes light red, bringing with it, in order, 1(4,8) and 1(5,5).”

*The bridge at work in a lite trap. Note how the SE 6 group carries the red lite tree. Oops, my bad. I didn’t mark the c8 6 slink.*

Now E1 => C1, for:

““The red-blue trap kills 4(5,5) and 1(8,5), establishing the 1-clues and expanding green-blue.”

*Adding a lite blue tree from 5r8c1, blue light 4r8c9 and green 4 trap 4r8c7 to leave full blue 4r8c9, green 5r8c9 and blue 5c7c9.*

As 5r8c6 is trapped, both 5’s in the South box become blue lite and blue is wrapped. Does that have consequence for red/orange? Yes, if orange is true, orange 8r6c2 sees green 8r6c5, its last candidate. So orange is wrapped. We have the clues of the color trial, without the trial.

Deleting the blue and orange candidates and trees, and promoting the green and red trees to full colors, we carefully pick out the second right page trial free solution among the 12 ultrahardcore review puzzles.

Deleting the blue and orange candidates and trees, and promoting the green and red trees to full colors, we carefully pick out the second right page trial free solution among the 12 ultrahardcore review puzzles.

Next time, we’re staying with this puzzle UHC 311 to revisit its AIC Building phase, and to show how AIC building can be organized for systematic, exhaustive exploration, without the human-like solvers.

And for the readers that have been looking at my early posts on Susets(10/25/11), and ALS Toxic Sets(7/17,25/12), I’ll soon be updating and applying susets to generate the ALS map, and to duplicate by hand the review’s impressive solver ALS findings of 11/3/20 systematically and exhaustively.

The first objective was the cluster merge. Coloring gives you a color bridge, when green and red in the Northwest box declares that green and red cannot both be true, so either blue or orange is true, if not both. Now any candidate seeing both is false.

But lite coloring does more. Red 3r2c3 winks at green 3r3c2, which slinks with any blue, by definition. So if red is true, blue is true – a coloring cluster merge.

The merge follow up

allows a 1-chain ANL, expanding the cluster.

Let’s walk through the added lite trees. From now blue 1r4c9, wink to 1r1c9 and slink to 7r1c9 for the first arrow, then wink to 7r2c7 and slink to 1r2c7, then wink to 1r8c7 and slink to 6r8c7. Combine that with the tree from green 3r2c5 to 6r4c4, and we trap 6 r4c7.

Continue from 6r8c7 with wink to 6r8c5 and slink to 6r7c5. Add the arrow from green 7r3c5. 6r7c5 is green lite and blue lite. It’s a lite coloring confirmation!

Following up with S6 => SE47, there’s an expansion trap on 1r4c1, and a tidy ALS_61 with a Cc5 boxline taking advantage

We add a green lite tree, then realize we can extend the blue lite tree to 6r1c6, doubling up with the green lite tree already there. Now 6r1c6 is true regardless of which tree becomes true. One does.

Now it gets a little hard to follow after

N6 =>(C6, N2, N4)

when ALS_54 overwrites the arrows. How do you spot such a thing. The X is easy enough, but the ALS skipping 4 and the 4 value group? No way.

The last 4 in c3 becomes SW4, and a naked triple 279 comes out.

And finally, after

a lite coloring wrap. If green is true, two green S7’s, and blue mops up.

To end lite coloring look backs, we go back to an Dov Mittelman email early in the ultrahardcore review, which he later had to explain to me, about his “light coloring”. Yes, I changed it to ‘lite’ after, rather than ‘light’ before. I also put in arrows, specifically representing wink to slink lite tree segments.

Go back to *A Coloring Trial Weakens UHC 311* of November 17, just before the coloring trial. You just might be able to step in with lite coloring, and avoid the trial, by getting a wrap. And if you’ve been hanging back, you just might be availing yourself of this crazy blog.

Here is the bypass just before the resolution on c4. Following the trace, note how the three 4-fills were resolved.

The basic trace includes a hidden single and hidden unique rectangle.

The hidden UR is signaled by the 35 naked pair and the perpendicular 5-slinks. Something must be going on. Then you see that if the slink partner 3r5c7 is true, 5 is forced into and 3 into the opposite corner for two solutions on the rectangle.

Then when you get to the harder fills, you notice the many line slinks on the bv values there. You throw the colors on, and there is a very subtle trap, and a boxline removal as well.

I’ll leave it there for you to explain, and slip the full disclosure in a little later.

I have Beeby along, and it finds the grouped nice loop, but removes the two 4’s only. In a nice loop, one end of link, strong or weak, is true. Beeby finds the other two later.

On the next grid, 3r1c3 is trapped in the NW box by the blue/green expansion. 2r2c2 is caught in the boyfriend trap. It shouldn’t be in a cell with a blue, when a green 2 can see it. Blue or green, it’s toast. Like earlier, when green made a hidden triple rejecting 9r2c7.

The X, XY AIC ANL works harder to remove another 2.

Now the easiest AIC of all, an XY ANL, makes a naked pair. Notice the blue/green expansions. Any lite trees yet?

In the South, the Naked Pair combines with another hidden unique rectangle and the blue green expansion to remove 3 candidates. A northerly Beeby ALS_41 creates a grouped pincer ANL.

Beeby’s ALS_38 collects the 8-candidate it missed in the nice loop.

Finally, the cluster hatches a lite tree. After Beeby’s grouped ALS_93 on the right and the ALS-wing on the left, we see a blue green expansion coming.

Now there’s room to show some lite trees. A fourth one combines blue lite 9r4c1 and green lite 9r8c2 to create a damaging naked pair.

That’s going to confirm 9r8c2 and 5r8c5. Careful, that doesn’t mean green is true. In fact, after

6r9c5 remains firmly connected in the blue/green cluster.

The follow up marking includes a green lite tree from 9r1c8, which finishes Only Extreme 347.

If another candidate in the starting cell r1c8 becomes green lite, the green is false. If green is true, there are two green candidates in the starting cell. Green can’t win for losing.

Next week, let’s trek back to another of my favorite collections, KrazyDad Insane, to take another look at Insane 425, (volume 4, book 2, number 5) in the post of July 23 2013. The issue is the two cluster coloring of the last grid of the post. In the next post of July 30 (one click reaches it) the two clusters are merged by pattern analysis. But you can ignore that. With the lite coloring reported so far, you can do the merge, and much more.

The KrazyDad Insane review was updated in December 2018. But it didn’t change the merge technique, so you can ignore that. What you shouldn’t ignore is coloring, which experts and coders have for too long after Andrew Stuart. Especially now, with its latest enhancement, lite coloring.

]]>The first lite coloring update is the post of November 28, 2017 on Castillo’s Only Extreme 261, which ended with a Not-Both trial.

One or both of the 7 terminals of the AIC is true. Also, one of them is green. For a while I used Not-Both trials to extend the slink network, the coloring. Here it would test of both 7’s being green. Here, the trial solved the puzzle.

An update post on Only Extreme 261 of April 30, 2019 showed that all of the 5 patterns leave a green 5 orphan, wrapping green and avoiding the coloring trial.

Without looking below, note your lite coloring on the grid above.

OK, on my lite coloring, the blue lite tree from 7r9c9 follows the path of the AIC, reaching 7r4c7, the uncolored terminal. If blue is true, 7r4c7 will be blue. That’s not enough to extend the cluster along the AIC, with green 7r4c9 and blue 7r4c7, but it’s similar progress with no trial.

And red is blue lite, everywhere, because if one red is true, all are. With green lite 8r8c8, that gives us a nice set to lite traps.

Now look at the lite tree from 7r1c8. 1r1c6 is blue lite. Using the grouped wink in the North box, 1r8c6 is blue lite. Using another grouped wink-slink, the r12c6 1-group is blue lite as well. Weird that any of these facts could be applied in a lite trap. But if green is confirmed, it’s all fake news.

Next week, we do a complete update of the 12/19/2017 post on Castillo Only Extreme 347. Originally it was a coloring trial used to illustrate the Sysudoku trial tracing technique. Phil Beeby’s solver finds simple chains to a color wrap with a lite coloring issue. There is also a bypass update featuring prominent use of 4-fills. Interested in trying it out along those lines? Don’t mind if you do.

Here’s the correction on a goof by yours truly. On the APE by Sudokuwiki and ALS_87 by Beeby, I deleted 7r3c8, instead of 7r2c8. In the post, that error is carried forward without incident, not affecting any eliminations or the coloring trial which solves 399.

Now coloring lite has a trap, on that very 7r3c8 spared by my error.

7r3c1 is blue lite, 7r6c8 is green lite.

Reader Dov Mittelman spotted a second lite trap, at 3r3c1, seeing blue lite 7 in r3c1 and green lite 3r6c1.

Beeby was enlisted, by stepping through its former finds and using its REMOVE feature, to take out the lite traps.

The 3r3c1 removal enables a finned swordfish, before the boomer from 7r6c8.

Also, a complex 1-way from 7r3c1, which sees the victim and starts the branch hacked slink chain to the other terminal of the ANL. True or not, it wipes out 7r3c5 and becomes the winning 7-candidate on r3.

After NW7, Beeby runs out of even the unlikely human options.

There are alternatives to the green coloring trial of the review. The green trial comes with the green lite trees. But the other options are advanced by the lite coloring.

Cc4 contains 6(1+8)(4+5), so orange comes with a C45 pair.

The SASdC in Nr1 ,

Nr1 = 5(3+9)(2+5) + 572 comes with the aqua candidates

Also there’s a pattern situation on the 7-panel left by my r3c8 lite removal.

The short dashed pattern reaching r1c2 is rejected by Dov’s lite trap, leaving the third tan/aqua cluster in the 7’s to be decided.

The point is that lite coloring enhances trials, when it fails to avoid them.

This time, compared to the green trial in the review, the green clues change little, because we have already incorporated most of their effects. The color expansion is in the red/orange cluster, which grows in a distinctive way, by adding naked pairs.

looks like I may have to try the red or orange army. Then I notice that in r2, orange will enable a now naked triple to confirm 4r2c5 and then red 1r2c1. No, Virginia, orange doesn’t get to confirm red. It gets tossed for trying, and collapse follows.

Compared to the review post, the color solution has the same reds, but a few more greens.

There’s now a lite coloring page in The Guide, mostly based on last week’s post. It’s a child of the coloring page (drag off right). Guide pages on Sysudoku trials are coming soon.

As to posts, the next one will look back to a previous attempt to extend coloring. Suggestions on collections to review or topics to explore are welcome, either in comments on posts or at sysudoku@gmail.com. Please attach comments to relevant posts.

]]>Although I can tell you how lite coloring works, I hadn’t been blessed with good examples until another reader friend Gordon Fick put me onto the “Red Russel special”. Here’s why examples are scarce: A lot of hard review puzzles finish with coloring, but it’s pick, pick , pick, candidate by candidate until a cluster or two can be started, and then it ends quickly. Lite coloring is needed most when the clusters are there but aren’t working.

The puzzle I name here the *Red Russel Special* is a Sudoku presented long ago on the EnjoySudoku forum by contributor “Red” Russell. Its givens and prospects for Sysudoku solvers appeared at the end of the previous post, the last one of 2020.

The basic trace shows where the four X-wings and hidden quad occur. Yes, nada on the bypass. Beginners, you can learn all about traces in The Guide, linked above.

Now as we stop on that line marking 4-wing, the grid shows why, as we mark the r8 slink in its southwest cell corners, it’s so easy to spot the matching slink in r3. Of course we need to leave the fish icons on to pick up more 4-wing victims as we line mark more rows.

Pardon me for dwelling so long on Basic Sysudoku, but the Red special illustrates line marking X-wing spotting so well. If you don’t actually do it this way, take the time to follow the line marking as line fill strings get longer and longer. We line mark the easiest first. If it’s confusing, then read the line marking page in the Guide along with the grid pictures.

I missed one X-wing at first, but here is all four of them together, the first two on rows. and the second two on columns.

Yes, Virginia, X-wings are fish. For what makes a fish a fish, go through Advanced to fish in the Guide.

Let’s leave the hidden quad for later, and recognize that each X-wing is a coloring cluster that could grow. Coloring? Better see

Guide/Advanced/ Coloring

The clusters are small and bear no apparent relation to each other. This “Red “ Special shows how lite coloring connects the clusters, extending their capability to trap and remove outside candidates.

Here is a definition of lite coloring:

The definition itself explains how lite color extensions create new traps. If an outside candidate sees two opposing cluster colors, lite or regular, it is seeing a true candidate, because all candidates of a true color are true.

The definition doesn’t tell us how to identify the lite color candidates. They come in a sequence. With any color candidate, full or lite, identify slink partners seen by that candidate. They are false when the starting candidate is true. And in each case, the slink partner is color lite in the starting color. The color lite procedure is, generate a branching tree of lite candidates of the starting color. Naturally, the root of the tree is a full color candidate. The sequence does not include candidates of the opposing color, which generate branching sequences of lite candidates of the opposing color.

Let’s walk through the marking of a tree of lite candidates of the Red Special clusters.

Starting with blue 4r3c1. If true it erases 5, 6, and 7, and no other 4’s, except the green ones. Only 6 is a slink partner, and its slink partner is blue lite. It is blue when blue is known to be true. The cell corner slink marking directly assists lite marking.

In Sysudoku, we draw arrows to mark lite candidates. The arrow stands for a wink to a candidate erased by blue and slink to the lite blue candidate. You can use either the arrow or the wink/slink sequence for marking. It doesn’t matter which blue candidate the arrow comes from. Any candidate on any arrow path from a blue candidate is blue lite. What matters is blue becoming true.

Lite coloring inventor Mittelman uses lighter shades of the same color for lite candidates, but I use arrows. For one reason, a candidate can be lite in more than one color. Also, in PowerPoint, my colors are already light shaded to make values visible, so it’s hard to make them lighter.

Continuing the lite sequence from blue 4r3c1, it’s OK for blue lite 8 to claim yellow r5c2 is also blue lite. But that doesn’t mean yellow is blue. It becomes blue only when blue is confirmed to be true.

The other blue candidate 4r8c5 claims red 6r7c9 as blue lite! A red candidate can act as blue lite in a trap. It does that, making 4r9c9 blue lite. Since 1r9c1 is green lite, 4r9c3 which sees blue lite and green lite in r9, is false. Red 6r7c9 is red, but blue lite 4r9c9 is not necessarily blue, but for sure, blue or green is true, so blue lite 4r9c3 or green lite 4r9c3 is true. The trap works.

But there is something else. If 3r7c5 is confirmed, that forces an orange candidate to be false . But orange and red are opposing cluster colors. If any orange candidate is false, they all are, and all red candidates are true. So now, red is true when blue is true. How about that? Red is blue. A merge of clusters.

We don’t see it here, but a lite coloring wrap occurs when lite coloring chains from opposing cluster colors meet on a candidate of another full color. The full color is true, and its opposite is false. No waiting for a chain color to become confirmed.

Let’s merge the red/orange cluster with the blue/green cluster and re-mark the red/orange lite trees as new blue/green lite trees.

1r3c6 is blue lite and green lite. It wins either way .

6r1c7 sees green in the NE box and lite blue 6r1c1.

3r6c7 sees green lite 3r1c7 in c7 and blue lite in the East box.

It’s a short follow up to blue 3r7c5 and a green wrap.

Now green lite sequences disappear, and blue lites become blue clues. Including yellow 7r5c2. Yellow 7’s are confirmed as blue, and the purple 7’s they see are removed.

The investment in lite coloring pays off with the additional blue clues, and collapse is immediate. Just be careful picking through the clutter.

Here’s the lite coloring solution with the surviving blue color.

Way back on the Basic trace, the hidden quad appears as X-wings and removals are removed, on the closing scan of c5. The subset candidates are encircled in light blue and the blue diamonds are the removals.

One more line marking fact: although all the candidates of the quad are identified when r5 is marked, the columns have not been examined for subsets. Rows or columns, whatever set finishes first, the unmarked “other” lines are examined for slinks and subsets, i.e. they are “closed”.

The quad removals bring a sharp collapse. Here is the Sysudoku trace, down roughly to the green wrap above.

The Red Russel Special gained attention for several reasons. It’s posted here, and in the Guide, because it illustrates so well the technique and possible results of Dove Mittelman’s lite coloring.

Next week, we correct an error in the December 8, 2020 post and make another attempt at a trial free solution to ultrahardcore 399 with lite coloring. The 7 in r2c8 belongs in r3c8. I’ll leave it uncorrected for now. Aw go ahead and DIY it, you’ll see where the error applies.

In line marking UHC 487, r6 gives you a naked single W7, free of charge.

Then in r1, there it is, another skyscraper in the 2’s. This time, the ANL victims are slink partners. We get two clues, and we don’t need to leave the chain behind.

But that’s not all. The S2 clue leaves a 6-slink in r9 that matches a slink partner in r7. This skyscraper we leave on for another victim in r4.

After this promising start, my solver team gets nothing, except a truly impossible Beeby complex 1-way for one more removal, plus a SEc8 boxline removal.

Readers who despair over my diagrams should see the corresponding Beeby chain notation below.

The special characters mark the departure of winks to erase an interfering candidate. The symbol repeats at the slink created by the erasure. These erasures are valid only in the chosen direction of the chain.

To avoid unnecessary notation reading headaches, ignore the special characters at first, simply accepting the (=) slinks. *Then* draw the branches from the character marked candidate whose erasure enables the slink started at the second symbol.

On the 6-panel, there are only two South to North freeforms containing 6r9c1. Starting the trial on the first 5 of 7 candidates, the trial itself will probably select the right path, if there is one. If not, then the failure confirms 6r9c8 as a clue, removing it from the 6-panel.

The trace is long, but there is a steep collapse to a contradiction as two 3’s are forced inton c7. The clue and two removals do not wake up the solvers.

Next I look at other edges. From East to West, there is only one pattern containing 6r3c9. If the freeform fails, 6r4c7 is an orphan as well, because c9 must be included.

Again this test of 6 candidates fails quickly.

Now we have two sets of East to West freeforms, three from r4c9 and and two from r6c9. The red one fails to reach c1.

In a trial of patterns containing r4c9, an initial coloring yields two traps.

Then

gets to a 497-wing

followed by an XY ANL

and a green wrap.

So what pattern was the true one? Now we know.

That winds up our review of Stefan Heine’s ultrahardcore right page collection. Let’s keep trying for ‘no trial’ humanly practical solutions to these gems.

So what happens now?

There being no urgent reader requests for another collection review, early 2021 posts will be about expansions and revisions of *The Guide*, that link on the bar menu. You will be in on explorations, independent of a review schedule, of current and prospective Sysudoku Guide topics, prior to its publication in book formats.

Posts will continue to be weekly, Tuesday noon EST, but I expect the blog will be much more amenable to meaty comments and very careful replies. As many readers know, I keep comments of interest only to myself to myself. Comment about your ideas. Put them on a post or a page.

The first 2021 Guide topic is *lite coloring*, to my knowledge, an invention of reader Dov Mittelman.

Thanks to an alert from another expert friend Gordon Fick, here’s a puzzle recently highlighted on the EnjoySudoku forum that provides an excellent example, when you overlook the hidden quad that destroys it. Next week, you’ll get my definition of lite coloring. If you’re into Sudoku Basic, try to corral three X-wings in line marking, then the quad. The puzzle is attributed to “Red”, a.k.a. Ed Russell, an early regular forum contributor. The forum offers “Pattern Game” solutions for the Ed Russell special, which could be one of those Guide topics later.

The hidden unique rectangle left on grid is signaled by the bv 6 and 7 repeated in three more corners. The resolution method is marked by the three side slinks in the UR value 6. If true, 7r8c9 generates a reversable rectangle in a double solution.

Both solvers now quiet, my choice for a trial is the Single Alternate Sue de Coq Cr5, with contents described by 7(5+9)(1+8) +817.

That’s because it can’t contain 5 and 9. It must be 5 or 9 or neither. We test neither, i.e. 817.

The test reaches a contradiction on the 9^{th} breadth first level, leaving the Sue de Coq

Cr5 = 7(5+9)(1+8).

Now the (5+9) and the bv 59 sweep other 5’s and 9’s from r5, and the 9r5c9 removal brings the boxline removing 9r3c7.

This leaves Sue tugging on our sleaves with two Single Alternate SdC’s

Er4 =1(2+8)(7+9)+971 and

Wr4 = 5(7+9)(2+8)+582.

Both Er4 =971 and Wr4 =582 produce the same contradiction as before, and both remove 7r4c8, leaving the pair E28 to remove 8r6c8.

After a 2-chain ANL, Beeby does a pair of ALS-wing, the first, depending on singles values,

The second, sharing an ALS with the first. The 7r4c2 removal allows

a finned 7-wing. The 7r7c9 fin sees the victims, so if it’s true, they’re toast and if it’s false, the 7-wing toasts them.

Next, after a 7-chain ANL,

and NE3 => N3,

Beeby’s next overlapped ALS-wing.

We sit back and watch as r5 naked triple generates W1 and the West box naked quad.

Now what is that thing?

It’s a classic boomerang with an ALS providing the closing slink.

It’s also a 1-way. If 8r7c4 is true, 2r7c4 is not, and if 8r7c4 is false, the ALS AIC promotes another 2 in the South box.

The removal adds a critical slink for the hidden unique rectangle.

Seeing an XY chain,

And an easy AIC, I think that closing this down with coloring may be overdue.

On first coloring, the bridge not(green and orange) becomes mute as green is wrapped by a trap induced boxline.

In the follow up and red/orange expansion,

1, 2 and 4 squeeze out 7 and 9 and the Sc5 boxline and trap removing 7r8c4 confirms orange.

Here’s the colored solution.

One more ultrahardcore to do, UHC 487. The site and the portable versions of the Guide need attention, and the only reason for doing another collection is to confirm any reader opinion that it is likely to reveal more about human solving. Stefan Heine’s ultrahardcores has certainly done that.